(x^2+1)^2-6(x^2+1)+9=0

Jasonbradley

2 min read

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Jun 17, 2024

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Solving the Equation (x^2 + 1)^2 - 6(x^2 + 1) + 9 = 0

This equation might look intimidating at first glance, but we can solve it using a simple substitution.

Substitution Method

1. Let's make a substitution: Let y = x² + 1. This will simplify the equation.

2. Substitute: Now, our equation becomes: y² - 6y + 9 = 0

3. Solve for y: This is a quadratic equation that can be easily solved by factoring: (y - 3)(y - 3) = 0 Therefore, y = 3

4. Substitute back: Now, replace y with x² + 1: x² + 1 = 3

5. Solve for x:
   x² = 2 x = ±√2

The Solutions

Therefore, the solutions to the equation (x² + 1)² - 6(x² + 1) + 9 = 0 are:

x = √2 and x = -√2

Conclusion

By using a simple substitution, we were able to transform a seemingly complex equation into a familiar quadratic equation. This demonstrates the power of algebraic manipulation in simplifying mathematical problems.